Finite Group Databasebeta

← #241316

Subgroups — #241316

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic
2#21964C212yesmaximal
3#20797C211yesmaximal, characteristic
4#10577C212yesmaximal, complemented
5#10576C212yesmaximal, complemented
6#10574C212yesmaximal, complemented
7#10573C212yesmaximal, complemented
8#8041C211yesmaximal, characteristic
9#7897C212yesmaximal
10C2×#427—22no—
11C2×#423C2212yes—
12C22×C4×C8C2211yescharacteristic
13#2844—22no—
14#2628—24no—
15#2628—22no—
16#2571—22no—
17#2571C2212yes—
18#2551—22no—
19#2551C2212yes—
20#2285C2211yescharacteristic
21#2285C2212yes—
22#2285C2211yescharacteristic
23#1948—22no—
24#1948C2214yes—
25#1947—22no—
26#1947C2214yes—
27#1846—22no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1844C2211yescharacteristic
31#1844C2211yescharacteristic
32#1809—24no—
33#1296C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C22×M16—22no—
38C23×C8—21no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—24no—
41C2×#117—22no—
42C2×#117—21no—
43C2×#117D412yes—
44C22×C42C2311yescharacteristic
45#427—48no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—24no—
49C2×#106—22no—
50C2×#106—22no—
51C2×#106—24no—
52#423—24no—
53#423Q818yescomplemented
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2314yes—
56C2×C4×C8—21no—
57C2×C4×C8—22no—
58C2×C4×C8Q814yescomplemented
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#400—28no—
62#400—24no—
63#399—24no—
64#395—28no—
65#395—24no—
66#393—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#384D412yes—
71#384D412yes—
72#384C2311yescharacteristic
73#384C2311yescharacteristic
74#384C2311yescharacteristic
75#384C2311yescharacteristic
76#382—22no—
77#382—24no—
78#382—22no—
79#382—21no—
80#382D412yes—
81#380—22no—
82#380—22no—
83#380—22no—
84#380D414yes—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2314yes—
89#340—24no—
90#340—24no—
91C22×Q8—24no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C2×M16—28no—
95C2×M16—82no—
96C2×M16—82no—
97C22×C8—24no—
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—24no—
103C4⋊Q8—84no—
104C4⋊Q8—84no—
105C4⋊Q8—84no—
106#117—44no—
107#117—44no—
108#117—82no—
109#117—82no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—41no—
129C2×C42—22no—
130C2×C42—24no—
131C2×C42—41no—
132C2×C42C2×D412yes—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×Q812yes—
135C2×C42C2×Q814yes—
136C2×C42#4112yes—
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—44no—
140#107—84no—
141#106—44no—
142#106—44no—
143#106—84no—
144C4⋊C8—28no—
145C4⋊C8—24no—
146C4⋊C8C2×Q818yes—
147C4×C8—24no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8C2×Q812yes—
151C4×C8C2×Q812yes—
152#95—24no—
153#95—24no—
154#95—28no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—42no—
159C24#12011yescharacteristic
160C2×Q8—88no—
161C2×Q8—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4#12012yes—
175C22×C4#12011yescharacteristic
176C22×C4C22⋊Q812yes—
177C22×C4C22⋊Q814yes—
178C22×C4C4⋊Q812yes—
179C22×C4C4⋊Q814yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D412yes—
182M16—88no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—216no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—28no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—88no—
214C4⋊C4—84no—
215C22⋊C4—82no—
216C22⋊C4—84no—
217C22⋊C4—82no—
218C42—44no—
219C42—44no—
220C42—44no—
221C42—82no—
222C42C22⋊Q812yes—
223C42C22⋊Q814yes—
224C42C4⋊Q812yes—
225C42C4⋊Q814yes—
226C42C22×Q811yescharacteristic
227C42C22×Q812yes—
228C42C2×#4111yescharacteristic
229C23—22no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23#39512yes—
234C23C2×#12011yescharacteristic
235Q8—168no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#39514yes—
261C2×C4#39512yes—
262C2×C4#45312yes—
263C2×C4#45612yes—
264C2×C4C2×C22⋊Q811yescharacteristic
265C2×C4C2×C22⋊Q812yes—
266C2×C4C2×C4⋊Q811yescharacteristic
267C2×C4C2×C4⋊Q812yes—
268C2×C4#57612yes—
269C2×C4#57712yes—
270C2×C4#57712yes—
271C2×C4#57812yes—
272C8—24no—
273C8—24no—
274C8—48no—
275C22—24no—
276C22—22no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22C2×#39511yescharacteristic
281C22#301212yes—
282C22#301312yes—
283C22C2×#45311yescharacteristic
284C22C2×#45611yescharacteristic
285C4—24no—
286C4—21no—
287C4—21no—
288C4—82no—
289C4—84no—
290C4—84no—
291C4—82no—
292C4—161no—
293C4—161no—
294C4#206014yes—
295C4#308112yes—
296C4#308212yes—
297C4#316312yes—
298C4#316412yes—
299C2—42no—
300C2#1868611yescharacteristic
301C2#1868811yescharacteristic
302C2#1870212yes—
303C2#2355712yes—
304C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10577C2semidirect
5. #10576C2semidirect
6. #10574C2semidirect
7. #10573C2semidirect
53. #423Q8semidirect
58. C2×C4×C8Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#376422
#219642
#207972
#105772
#105762
#105742
#105732
#80412
#78972